Effect of boundary conditions on the dynamic instability and non-linear response of rectangular plates, Part I: Theory

Abstract

The dynamic stability and non-linear response of parametrically excited rectangular plates are theoretically analyzed under four different sets of boundary conditions. By applying the approach of generalized double Fourier series to the dynamic analog of the von Kármán equations, the problem is reduced to that of the parametric vibration of a finite-degree-of-freedom system. The first order generalized asymptotic method is used to solve the non-linear temporal equations of motion, and only principal parametric and combination resonances are considered and investigated in this work. Calculations are carried out for rectangular plates of various aspect ratios under each case of boundary conditions. Regions of parametric instability, as well as frequency response curves corresponding to principal parametric resonances associated with the lower mode shapes, together with combination resonance of the sum type, are determined for different values of various system parameters. The results obtained indicate that the boundary conditions play a crucial role in determining the number of possible resonances, and the stability and response characteristics of rectangular plates.